Homotopy equivalences on 3-manifolds

Abstract

ABSTRACT. Suppose M3 is a 3-manifold and /: M3 -* X is a homotopy equivalence onto an ANR X. In this paper the cellularity properties of point preimages under / are studied.It is shown that for every open cover a of X there exists an open cover ß of X such that if / is a /3-equivalence then each /_1(i) is a-cellular in M^ X R1. In fact, the (open) cellularity occurs in a continuous fashion and so the map / can be approximated by a Euclidean bundle map.0. Introduction. Suppose Mn is an n-manifold, X is an ANR and /: M" -► X is a map. The questions studied in this paper concern the relationship between homotopy properties of / and the structure of point inverses under /, particularly in dimension n = 3.The case in which / is a cell-like map (or equivalently, / is a fine homotopy equivalence [3]) is well understood. Each point inverse is a cell-like set and McMillan [5] has given a criterion for determining whether or not these cell-like sets are actually cellular as long as n > 5. Freedman [2] has recently extended that result to the case n -4. It follows that for every value of n, f is cell-like if and only if each /-1(z) is cellular in M+ x R1. (Here M+ denotes M plus a collar attached along the boundary of M.) In addition, Rushing [7] has shown that if n > 5, then / can always be approximated by an (n + l)-disk bundle map .In [6], Montejano and Rushing generalize the preceding results to the case in which / is an a-equivalence.Their theorems are true generalizations since a celllike map of ANR's is an a-equivalence for every open cover a of the range while not every a-equivalence is a cell-like map. The proofs given by Montejano and Rushing are valid only in dimensions > 4, and it is the purpose of this paper to prove the low-dimensional cases of their theorems. Since the case n < 2 is trivial, that means that we concentrate on the case n -3.Before stating the main theorem, we give a definition. DEFINITION [6]. Let E C M" be a closed subset of an n-manifold M, let B be a metric space and let a be an open cover of B. A map p: E -> B is said to be a-cellular (in Mn) if for every b G B and e > 0, there exists an n-cell D C Mn such that DDE C p~1(Na(b)) and p-\b) clntflcflC Ne{p-\Na{b))) C Mn.Notice that /: Mn -► X is cellular if and only if it is a-cellular for every open cover a of X.